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Future asymptotics of vacuum Bianchi type
V I0 solutions
J. M. Heinzle and H. Ringstrom
REPORT No. 29, 2008/2009, fall
ISSN 1103-467XISRN IML-R- -29-08/09- -SE+fall
Future asymptotics of vacuum Bianchi type VI0solutions
J. Mark Heinzle∗
Gravitational Physics, Faculty of Physics,University of Vienna, A-1090 Vienna, Austria
andMittag-Leffler Institute of the Royal Swedish Academy of Sciences
S-18260 Djursholm, Sweden
Hans Ringstrom†
Department of Mathematics,Royal Institute of Technology,S-10044 Stockholm, Sweden
Abstract
In this paper, we present a thorough analysis of the future asymptotic dynamics of spatiallyhomogeneous cosmological models of Bianchi type VI0. Each of these models converges to aflat Kasner solution (Taub solution) for late times; we give detailed asymptotic expansionsdescribing this convergence. In particular, we prove that the future asymptotics of Bianchitype VI0 solutions cannot be approximated in any way by Bianchi type II solutions, whichis in contrast to Bianchi type VIII and IX models (in the direction toward the singularity).The paper contains an extensive introduction where we put the results into a broader context.The core of these considerations consists in the fact that there exist regions in the phasespace of Bianchi type VIII models where solutions can be approximated, to a high degree ofaccuracy, by type VI0 solutions. The behavior of solutions in these regions is essential forthe question of ‘locality’, i.e., whether particle horizons form or not. Since Bianchi type VIIImodels are conjectured to be important role models for generic cosmological singularities, ourunderstanding of Bianchi type VI0 dynamics might thus be crucial to help to shed some lighton the important question of whether to expect generic singularities to be local or not.
∗Electronic address: [email protected]†Electronic address: [email protected]
1
1 INTRODUCTION 2
1 Introduction
In order to put the results of the present paper into a broader context, let us begin by quoting apassage from Misner [11] concerning Robertson-Walker spacetimes:
. . . if the 3◦K background radiation were last scattered at a redshift z = 7, then theradiation coming to us from two directions in the sky separated by more than about30◦ was last scattered by regions of plasma whose prior histories had no causal relation-ship. [. . . ] Robertson-Walker models therefore give no insight into why the observedmicrowave radiation from widely different angles in the sky has very precisely (. 0.2%)the same temperature.
To resolve this ‘horizon problem’, Misner suggested to abandon Robertson-Walker models and toconsider spacetimes of Bianchi type IX instead, which were conjectured by Belinskiı, Khalatnikov,and Lifshitz [1, 10] and Misner and Chitre [4, 11] to be the paradigm in the understanding ofspacetimes with ‘generic singularities’. Misner argued in [11] that Bianchi type IX models mightexhibit a behavior toward the singularity which is quite different from that of Robertson-Walkermodels. Based on the absence of particle horizons in the x-direction in the flat Kasner solution(Taub solution)
−dt2 + t2dx2 + dy2 + dz2 (1)
on (0,∞)× T3, and the expectation that (generic) type IX solutions are recurrently well approxi-mated by this solution (and its equivalent representations arising from permuting the axes), Misnerconjectured that particle horizons should be absent in (generic) Bianchi type IX solutions and thusproposed a remedy for the horizon problem.
Our perspectives on the initial singularity of the Universe have changed quite drastically in thelast forty years. In particular, the introduction of the concept of inflation, which is a formidablecontestant to explain the causal structure of the early Universe, has brought about a return toFriedmann-Robertson-Walker models. Nevertheless, we consider the question of the causal struc-ture close to the singularity of more general spacetimes to be still one of fundamental importance,the main reason being that it plays a crucial role in the study of singularities in the spatiallyinhomogeneous setting—we will explain this in more detail below.
Particle horizons, PDE perspectives and heuristics. In order to illustrate that the questionof whether particle horizons form or not can be posed in a meaningful way in general classes ofspacetimes, let us consider the maximal globally hyperbolic development, say (M, g), of some initialdata, where M = (t−, t+) × Σ and Σ is the initial manifold. Let us, furthermore, assume that ∂tis future oriented timelike with respect to g, that {t} × Σ are spacelike Cauchy hypersurfaces fort ∈ (t−, t+), and that all causal geodesics are incomplete to the past. Then t = t− can be thoughtof as a singularity. Let γ be an inextendible causal curve in (M, g), and let At be the subset of Σcorresponding to the intersection J+(γ)∩ ({t}×Σ). Note that it is necessary to control the initialdata induced on this set in order to predict the behavior of the solution along the causal curve γall the way to the singularity. Various definitions of locality, phrased in terms of the properties ofAt as t → t−, are now concievable. If At = Σ for all t and all γ, then, clearly, it is reasonable tosay that the singularity is not local and that particle horizons do not form. Nevertheless, to onlyuse this as a criterion is somewhat too crude; in the case of (1) for instance, At is never all of T3,regardless of the choice of curve. On the other hand, in that case, At has non-trivial topology,and it is not contained in a subset of T3 homeomorphic to a 3-ball (cf. the notion of late timeobservers being oblivious to topology, introduced in [17] in the context of cosmological solutionswith accelerated expansion). This is an indication that, even though particle horizons form in thesense that At 6= T3, the formation is partial. If one considers (1) as a metric on (0,∞)×R3, thenAt is unbounded, and R3\At has non-trivial topology, again signalling a partial absence of locality.The above discussion indicates that various definitions are possible, but we will not attempt to givea formal definition in the above degree of generality. Moreover, we will, in practice, assume that wehave some preferred geometric structures (e.g., in the case of spatial homogeneity, a constant meancurvature foliation and a Gaussian time coordinate) which allow us, in an unambiguous way, to fix
1 INTRODUCTION 3
a Riemannian metric (or, possibly, a family of equivalent metrics) on Σ, with respect to which wecan measure the size of the set At. If the set At converges to a point for every inextendible causalcurve, as t → t−, one could then define the singularity to be local. This is the definition we willuse below.
The singularity theorems of Hawking and Penrose predict that there are, generically, singularitiesin the sense of causal geodesic incompleteness. However, these theorems do not provide anyinformation concerning the formation of particle horizons, nor on the issue of curvature blow up;for example, is the Kretschmann scalar (i.e., the Riemann curvature tensor contracted with itself)unbounded in the incomplete directions of causal geodesics? A weaker question to ask would be:Does strong cosmic censorship hold? To answer such questions, it seems necessary to carry outa detailed analysis, which would entail, in all probability, considering Einstein’s equations from aPDE point of view. When taking the PDE perspective, the causal structure is very important. Asan illustration, it is instructive to consider the proof of curvature blow up in the direction towardsthe singularity in spacetimes with T3-Gowdy symmetry. In that case, the analysis is based onthe fact that it is known a priori that particle horizons form, and it turns out to be crucial toconsider regions of the spacetime that are as small as possible given the constraint that they stillbe large enough to make it possible to predict the behavior along a fixed causal curve going intothe singularity (and that these regions shrink to zero in the approach to the singularity). For amore complete discussion, we refer the interested reader to [16], in particular the last section, andreferences cited therein. Needless to say, analyzing the causal structure will in general be part ofthe problem, i.e., it should not be expected to be possible to separate the problem of analyzingthe causal structure from the problem of analyzing the asymptotics of solutions.
So far, the difficulty of the problem has prevented a rigorous investigation of the structure of singu-larities of generic spacetimes; however, there have been a great many heuristic studies, pioneeredby the work of Belinskiı, Khalatnikov, and Lifshitz [1, 2, 10] and Misner and Chitre [4, 11]. Fromthese studies, a consistent picture emerges which describes generic singularities as spacelike, local,vacuum dominated, and oscillatory. In the Hamiltonian approach, the asymptotic behavior of thespacetime metric is described in terms of a ‘cosmological billiard’ motion in a (minisuper-)spacebounded by infinitely high walls (‘billiard’), see Damour, Henneaux, and Nicolai [5, 6] and refer-ences therein. In the dual dynamical systems approach, the asymptotic dynamics are representedby a system of ordinary differential equations, one for each spatial point, on the ‘silent boundary’,see [7, 18] and references therein. It is important to note, however, that these heuristic approachesto generic singularities crucially rely on the assumption of asymptotic silence and locality: Theassumption that particle horizons form guarantees localization in the spatial directions, which sug-gests that spatial derivatives in the PDEs are (generically) dominated by temporal derivatives, andthe equations result in an asymptotic system of ODEs [18].
In this manner, these heuristic approaches to generic singularities consistently suggest that theasymptotic dynamics of generic spacetimes are intimately connected with the asymptotic dynamicsof spatially homogeneous vacuum solutions (since these are described by ODEs that coincide withthe asymptotic ODEs of generic cosmologies); of particular relevance are the oscillatory asymptoticdynamics of the Bianchi models of type VI−1/9, VIII, and IX. It is therefore natural (and essen-tial) to consider these spatially homogeneous models and to investigate whether the singularitiesthese models form are local or not. (Note that if these singularities were not local, our heuristicconceptions of generic spacetimes would be utterly inconsistent.)
Bianchi class A. We will here restrict our attention to the Bianchi class A vacuum spacetimes.These spacetimes can be defined as the maximal globally hyperbolic developments of left invariantvacuum initial data on unimodular Lie groups. The corresponding metrics can be written in theform (4), where the different Bianchi class A types are characterized by the values of nα, α = 1, 2, 3,given by Table 1. Let us remind the reader that the Bianchi type IX solutions recollapse, i.e., theyare past and future causally geodesically incomplete, and thus have both a future and a past sin-gularity. However, all the other Bianchi class A solutions (except Minkowski space and quotientstherof, which we ignore from now on since they are non-singular) have the property that theyare future causally geodesically complete and past causally geodesically incomplete. Turning to
1 INTRODUCTION 4
the question of whether there is localization of the causal structure or not, it is natural to startby considering the solutions with an additional local rotational symmetry (LRS). Only Bianchitype I, II, VII0, VIII, and IX admit an additional such symmetry (in the case of Bianchi type I,an example of an LRS metric is given by (1)). The corresponding solutions do not have a localsingularity in the above sense (at least not w.r.t. the natural foliation given by the hypersurfacesof spatial homogeneity). However, they are special in many ways: They are non-generic in theirrespective classes, and they can be extended; in particular, they constitute examples of singularitiesthat are not curvature singularities; see, e.g., [12]. Excluding these examples, the remaining space-times of Bianchi type I, II, VI0, and VII0 have local singularities. What remains to be consideredis thus solutions of Bianchi type VIII and IX that are not locally rotationally symmetric.
For the sake of definiteness, let us consider the past singularity, and let us assume that the met-ric (4b) is defined on (t−, t+)×G, where G is the unimodular Lie group under consideration andt = t− corresponds to the past singularity. Particle horizons form and the singularity is local (withrespect to the canonical constant mean curvature foliation) if and only if
3∑
i=1
∫ t0
t−
1√gii
dt < ∞
for some t0 ∈ (t−, t+), where the gii are given in (4b). In the case of Bianchi type VIII and IX,this condition can be reformulated to
∫ t0
t−
(√|n1n2|+
√|n2n3|+
√|n3n1|)dt < ∞, (2)
cf. (6) below. When analyzing the asymptotics, it is convenient to Hubble-normalize the variables,cf. (7), i.e., one divides the traceless part of the second fundamental form and the variables nα,α = 1, 2, 3, by the Hubble scalar (which is minus one third of the mean curvature). We willdenote the thus normalized traceless part of the second fundamental form by (Σ1,Σ2,Σ3) andthe normalized version of the nα by Nα. Furthermore, it is convenient to introduce a new timecoordinate by carrying out an analogous normalization, cf. (8). These variables were introducedin [20], and describe the essential dynamics of all the Bianchi class A types; the Nα should bezero or non-zero and have signs according to Table 1. In this picture, the Kasner solutions, i.e.,the Bianchi type I solutions, are given by demanding that all the Nα be zero, and, due to theHamiltonian constraint, they constitute a circle of fixed points. On this circle, there are threepoints that are referred to as the special points or Taub points; they are the ones correspondingto the flat Kasner solution (1). Formulating the condition (2) in terms of the Hubble-normalizedvariables and Hubble-normalized time, one obtains, in the case of Bianchi type VIII and IX, thecondition ∫ τ0
−∞(√|N1N2|+
√|N2N3|+
√|N3N1|)dτ < ∞ . (3)
The paper [14] contains a proof of the statement that the integrand in (3), say I(τ), convergesto zero as τ → −∞ for solutions of Bianchi type IX. Although the argument has been slightlysimplified in [8], the proof of the fact that I converges to zero is rather intricate, and proving (3) canreasonably be expected to be much more difficult. In view of these difficulties, it seems reasonableto turn to numerical methods. Starting with seemingly arbitrary initial data, numerically solvingthe corresponding ODE indicates that I should converge to zero exponentially as τ → −∞, andthat the α-limit set should coincide with the part of the boundary of the phase space where theintegrand equals zero (let us call this set the attractor and denote it by A). Consequently, theredoes not seem to be a problem. However, one can prove that there are no solutions with thisbehavior; if the α-limit set coincides with A, then I cannot converge to zero exponentially. Thisindicates that the behavior of the solutions is quite subtle and that if one uses numerical techniques,one has to be very careful. One of the reasons why the behavior is so subtle is that the solutionis expected to return an infinite number of times to regions of the phase space which are close tothe special points that correspond to solutions of the form (1), cf. the above arguments by Misner.In fact, the expectation is that the solution will spend most of its time close to such points. Since
1 INTRODUCTION 5
I(τ) need not decay in the vicinity of these points (let alone decay exponentially), it is not so clearthat (3) holds.
Bianchi type VIII. Let us turn to Bianchi type VIII. Concerning this class of solutions, it is noteven known whether I converges to zero or not. Furthermore, the behavior in the case of Bianchitype VIII is in some respects worse. For Bianchi type IX, it is known not only that I convergesto zero, but also that the convergence is almost monotone in the sense that for any ǫ > 0, thereis a δ > 0 such that if I(τ) ≤ δ at τ = τ0, then I(τ) ≤ ǫ for all τ ≤ τ0, cf. [14, Corollary 15.3,p. 471]. Proposition 6.2 of [15, p. 3802] proves that the analogous statement cannot hold in thecase of Bianchi type VIII. The results of [13] imply that there is a sequence of times such that Iconverges to zero along it, but due to the absence of almost monotone convergence, this statementdoes not allow much in the way of conclusions. Proposition 6.2 of [15] is based on considerationsof the future asymptotics of Bianchi type VI0 solutions. Bianchi type VI0 solutions all have anω-limit set consisting of a fixed point in A; in fact, the ω-limit set always coincides with a specialpoint, see also Lemma 3.1. Perturbing such a solution into the Bianchi type VIII class and goingbackward in time consequently results in a proof of the fact that there cannot be almost monotoneconvergence to the attractor in the case of Bianchi type VIII. This behavior should be contrastedwith that of a Bianchi type VII0 solution for which the ω-limit set is not contained in A. Given aBianchi VII0 solution, with N1, N2 > 0 say, the ω-limit set is a point on a line of fixed points inthe Bianchi type VII0 class (each element of which, incidentally, corresponds to a solution of theform (1)). In other words, the integrand I, considered for a Bianchi VII0 solution, converges toa non-zero number to the future. Perturbing such a solution into the Bianchi type IX class andconsidering the behavior toward the past, the conclusion is that the integrand remains essentiallyconstant. In the case of Bianchi type IX, the regions close to the lines of fixed points of Bianchitype VII0 are the worst ones in that, there, it is hardest to prove decay of I going backward intime. Nevertheless, the above observation concerning Bianchi type VII0 indicates that not thatmuch growth can occur. In some sense, this is the basis for the proof of the fact that I convergesto zero in the direction toward the singularity given in [14].
To recapitulate the above discussion, it is clear that if one wants to prove that (3) holds, i.e., thatthe singularity is local, the solutions in Bianchi class A that are most problematic are the onesof type VIII; at this stage there does not even seem to be any strong reason to conjecture thatI converges to zero in that case. The problematic regions of the phase space (i.e., the regionswhere I can grow from being very small to being of a definite size) are close to the special pointson the Kasner circle where the solution can be approximated by the future asymptotics of aBianchi type VI0 solution. A first step, admittedly a small one, in analyzing the problematicregion consequently consists of carrying out a detailed analysis of the future asymptotic behaviorin Bianchi type VI0. This is the theme of the present paper.
Bianchi type VI0. Computing the ω-limit set of Bianchi type VI0 solutions is quite simple, see,e.g., Lemma 3.1, but for the purposes outlined above it is necessary to consider the behavior ingreater detail. In [20], the function
Z−1 =43Σ
2− + (N2 +N3)
2
−N2N3
was introduced; in the present paper we use an analogous function, ζ, see (20). (The variables inZ−1 are the variables of [20] which are somewhat different from the ones used in the present paper.)In [20], the source of this function is quoted as Bogoyavlensky, cf. [3, p. 63] (where the notation Fi
is used). This function is monotone for a Bianchi type VI0 solution, but on the ω-limit set, both thenumerator and the denominator equal zero. Consequently, our knowledge concerning the ω-limitset does not allow us to draw any conclusions concerning the limit of Z−1. Nevertheless, it is ofgreat interest to know what this limit is. The reason for this is that Z−1 serves as a measure of thedistance from Bianchi type II behavior: For a Bianchi type II solution, one of the variables N2, N3
is different from zero (and the other is zero), hence Z−1 = ∞. Therefore, Bianchi type VI0 solutions(or type VIII solutions for that matter) are close to a type II solution (and can be approximated bysuch a solution) if and only if Z−1 is large. In the present paper, we prove that Z−1 converges tozero as τ → ∞ for every Bianchi type VI0 solution, see Corollary 4.2. Consequently, even though
2 BASIC EQUATIONS 6
the ω-limit set is a special point on the Kasner circle, the behavior of a type VI0 solution is very farthat of type II solutions and cannot be approximated by (a sequence of) type II solutions. (Thisis in contrast to Bianchi type IX, where the asymptotics is in fact largely governed by sequences oftype II solutions). The function Z−1 serves as a quantitative measure of this discrepancy; in thispaper, see Section 5, we illustrate in detail in which sense type VI0 dynamics differ from type IIdynamics. It seems reasonable to claim that Z−1 is also an important quantity to keep track ofwhen studying Bianchi type VIII solutions in the direction of the singularity, since the problematicbehavior occurs when the solution is well approximated by a Bianchi type VI0 solution for whichZ−1 goes from zero to some finite positive value. Therefore, by analyzing the asymptotics ofBianchi type VI0 in great detail, we hope to develop some understanding for the problems thatcan arise in the case of Bianchi type VIII.
Outline. This paper is organized as follows: In Section 2, we introduce the equations for Bianchiclass A models in the Hubble-normalized dynamical systems approach. In Section 3, we specializeto Bianchi type VI0; we use adapted variables to represent the phase spaceBVI0 of type VI0 modelsand we introduce the monotone function ζ (≃ Z−1), which is central to our analysis. Following thesepreliminary observations, we present a detailed and thorough analysis of the future asymptoticsof Bianchi type VI0 vacuum models in Section 4. The analysis is not quite straightforward; it isnecessary to use methods tailored to the problem, since standard techniques from the theory ofdynamical system fail due to the fact that the special point on the Kasner circle is a center fixedpoint. The main result of the paper is presented as Proposition 4.5, which gives the details of thefuture asymptotics. Finally, in Section 5, we illustrate the differences between type VI0 dynamicsand type II dynamics. We use units such that c = 1 = 8πG, where c is the speed of light and Gthe gravitational constant.
2 Basic equations
For a spatially homogeneous vacuum spacetime of Bianchi class A, there exists a symmetry-adapted(co-)frame {ω1, ω2, ω3} satisfying
dω1 = −n1 ω2 ∧ ω3 , dω2 = −n2 ω
3 ∧ ω1 , dω3 = −n3 ω1 ∧ ω2 (4a)
such that the metric takes the form
4g = −dt⊗ dt+ g11(t) ω1 ⊗ ω1 + g22(t) ω
2 ⊗ ω2 + g33(t) ω3 ⊗ ω3 . (4b)
The different Bianchi types are characterized by different spatial structure constants n1, n2, n3;in Table 1 we list all Bianchi class A types. In the case of Bianchi type VI0, we can, without lossof generality (by permuting the axes), settle for n1 = +1, n2 = −1, n3 = 0. For a representationof (4) in terms of a coordinate basis, see [19].
Let kαβ denote the second fundamental form, associated with (4), of the SH hypersurfaces t = constand define
H = − 13 tr k and σα
β = −kαβ + 13 tr k δαβ = diag(σ1, σ2, σ3) . (5)
The quantity H (‘Hubble scalar’) is related to the expansion of the normal congruence of the SHhypersurfaces, i.e., d
√det g/dt = 3H
√det g. The tensor σαβ is tracefree, i.e., σ1 + σ2 + σ3 = 0; it
can be interpreted as the shear. Furthermore, define
n1(t) := n1g11√det g
, n2(t) := n2g22√det g
, n3(t) := n3g33√det g
. (6)
Given the ‘metric variables’, i.e., (gαβ , kαβ), one can construct the ‘orthonormal frame variables’,i.e., (H,σ1, σ2, σ3, n1, n2, n3), and given the orthonormal frame variables, one can construct themetric variables (though this construction requires carrying out integrations of the shear variablesfor some of the Bianchi types). It is always understood that σ1 + σ2 + σ3 = 0.
2 BASIC EQUATIONS 7
Bianchi type nα nβ nγ
I 0 0 0II + 0 0VI0 + − 0VII0 + + 0VIII + − +IX + + +
Table 1: The Bianchi class A types are characterized by different signs of the structure constants(nα, nβ, nγ), where (αβγ) is any permutation of (123). In addition to the above representations,there are equivalent representations associated with an overall change of sign of the structureconstants; e.g., another type VI0 representation is (− + 0).
In the Hubble-normalized dynamical systems approach we define dimensionless orthonormal framevariables according to
(Σ1,Σ2,Σ3, N1, N2, N3
)=
1
H
(σ1, σ2, σ3, n1, n2, n3) . (7)
In addition we introduce a new dimensionless time variable τ , where
d
dτ= H−1 d
dt; (8)
henceforth, a prime ′ denotes the derivative w.r.t. τ .
Remark. The variables (7) are well-defined except in the case of Bianchi type IX. The reason isthat the Gauss constraint
6H2 = σ21 + σ2
2 + σ23 +
12 (n
21 + n2
2 + n23)− n1n2 − n2n3 − n3n1
guarantees that H remains positive if it is positive initially (with the exception of Bianchi type IX).In the case of Bianchi type IX, the variables cover half of the spacetime. In particular, Bianchimodels (except those of type IX) that are expanding initially are expanding for all times.
The Einstein equations result in a system of differential equations for the Hubble-normalized vari-ables; this system can be written, cf. [20],
Σ′α = −2(1− Σ2)Σα − 3Sα , α = 1, 2, 3 (9a)
N ′α = 2(Σ2 +Σα)Nα α = 1, 2, 3 (no sum over α) , (9b)
where
Σ2 = 16 (Σ
21 +Σ2
2 +Σ23) , (10a)
3Sα = 13
[Nα(2Nα −Nβ −Nγ)− (Nβ −Nγ)
2], (αβγ) ∈ {(123), (231), (312)} . (10b)
Apart from the trivial constraint Σ1 +Σ2 +Σ3 = 0, there exists the Gauss constraint
Σ2 + 112
[N2
1 +N22 +N2
3 − 2N1N2 − 2N2N3 − 2N3N1
]= 1 . (11)
Accordingly, the reduced state space is given as the space of all (N1, N2, N3,Σ1,Σ2,Σ3) suchthat (11) holds. The existence interval for solutions to (9)–(11) is (−∞,∞), with the exception ofBianchi type IX, in which case the existence interval is of the form (−∞, τ0) for some τ0 < ∞, see,e.g., [14]. Since Σ1 +Σ2 + Σ3 = 0, the dimensionless state space of the Bianchi type VI0 vacuummodels is 3-dimensional.
From these equations, the metric (4b) can be reconstructed by carrying out appropriate inte-grations. The pertinent equations are g′ii = 2(1 + Σi)gii (with i = 1, 2, 3) and the equationH ′ = −(1 + 2Σ2)H to recover cosmological time via (8).
3 ADAPTING TO BIANCHI TYPE VI0 8
3 Adapting to Bianchi type VI0
In Bianchi type VI0, the permutation symmetry of the three spatial axes (exhibited by, e.g., type Iand type IX models) is broken; the third axis is singled out. It is suggestive to globally solveΣ1 +Σ2 +Σ3 = 0 by introducing variables according to
Σ+ =Σ1 +Σ2
2= −Σ3
2, Σ− = −Σ1 − Σ2
2√3
, (12)
which yields Σ2 = Σ2+ +Σ2
−. Likewise, we adapt to the constraint (11) by defining
N+ =N1 +N2
2√3
, N− =N1 −N2
2√3
. (13)
In these variables, the constraint (11) reads
Σ2+ +Σ2
− +N2− = 1 . (14)
The constraint (14) can be employed to globally solve for N−; since N− is necessarily positive, wesimply obtain
N− =√1− Σ2
+ − Σ2− . (15)
The range of the variable N+ is restricted by the conditions N1 > 0 and N2 < 0; we obtain−N− < N+ < N−. Accordingly, in terms of the variables (Σ+,Σ−, N+), the Bianchi type VI0vacuum state space becomes an open ball, i.e.,
BVI0 ={(
Σ+,Σ−, N+
) ∣∣Σ2+ +Σ2
− +N2+ < 1
}. (16)
The system of equations (9) takes the form
Σ′+ = −2
(1 + Σ+
)(1− Σ2
+ − Σ2−)
(17a)
Σ′− = −2
[Σ−(1− Σ2
+ − Σ2−)−
√3N+
√1− Σ2
+ − Σ2−
](17b)
N ′+ = 2
[N+Σ+ +N+(Σ
2+ +Σ2
−)−√3Σ−
√1− Σ2
+ − Σ2−
](17c)
on BVI0 . This system possesses a regular extension to the boundary ∂BVI0 , which is the 2-sphereΣ2
+ +Σ2− +N2
+ = 1; see Figure 1.
The induced system on ∂BVI0 is associated with Bianchi types I and II. The equator of ∂BVI0 ,i.e.,
K# ={(
Σ+,Σ−, N+
)∈ ∂BVI0
∣∣N+ = 0, Σ2+ +Σ2
− = 1}, (18)
is a circle of fixed points, the Kasner circle. Each fixed point on K# corresponds to a Kasner metric(Bianchi type I vacuum metric) [19]. On K# there exist points that are associated with locallyrotationally symmetric solutions: Q1, Q2, Q3 and T1, T2, T3, see Figure 2. The ‘Taub point’ T3
is of particular importance for our purposes; it is associated with the flat Taub solution (Bianchitype I representation of a part of Minkowski spacetime, cf. (1)). We refer to [19] for details.
The two hemispheres of ∂BVI0 correspond to two different (but equivalent) representations of theBianchi type II vacuum state space. The northern (+) and southern (−) hemispheres, B± ={(Σ+,Σ−, N+) ∈ ∂BVI0
∣∣±N+ > 0}, correspond to N1 > 0, N2 = 0 (and N3 = 0), and N1 = 0,
N2 < 0 (and N3 = 0) respectively. The orbits of the dynamical system (17) on B+ and on B−correspond to Bianchi type II vacuum solutions. These orbits form a family of straight lines whenprojected onto (Σ+,Σ−)-space, see Figure 2.
Lemma 3.1. Let γ be an orbit in BVI0 . The α-limit set of γ is one of the Kasner fixed pointswith Σ+ > 1
2 . (Conversely, each of these fixed points is the α-limit for a one-parameter familyof orbits.) The ω-limit set of γ is the special point (Taub point) T3 (where Σ+ = −1, Σ− = 0,N+ = 0).
3 ADAPTING TO BIANCHI TYPE VI0 9
Σ+
Σ−
N+
Figure 1: The state space BVI0 . The equator of ∂BVI0 , N+ = 0, corresponds to the Kasner circle.The northern hemisphere, B+, which is given by N+ > 0, is a representation of the type II statespace; the same is true for the southern hemisphere, B−, given by N+ < 0. The orbit through thecenter is Γ.
T3
T1
T2
Q3
Q1
Q2
T3
T1
T2
Q3
Q1
Q2
Figure 2: Projection of the type II orbits of the hemispheres B+ and B− onto (Σ+,Σ−)-space. Thehorizontal axis is the Σ+ axis, the vertical axis is the Σ− axis.
Proof. Eq. (17a) implies that Σ+ is a strictly monotonically decreasing function on BVI0\K#.Application of the monotonicity principle, see, e.g., [19], yields the possible α- and ω-limit sets oforbits in BVI0 : α-limit points must be contained on K#\{T3}, ω-limit points on K#\{Q3}. Thelocal analysis of the fixed points1 on K# restricts the possibilities further and leads directly to thestatement of lemma.
There exists one orbit of (17) that is central to our considerations:
Γ : N+ ≡ 0 , Σ− ≡ 0 . (19)
Along this orbit, the system (17) reduces to the equation Σ′+ = −2(1 + Σ+)
2(1− Σ+), which can
be solved implicitly.
1Strictly speaking, the local analysis of the fixed points on K# has to be performed using a set of variables inwhich the dynamical system is C1 on the closure of the state space; the system (9) is well suited.
4 FUTURE ASYMPTOTICS OF BIANCHI TYPE VI0 VACUUM MODELS 10
Consider the function
ζ = −1− (Σ1 − Σ2)2 + (N1 −N2)
2
4N1N2=
N2+ +Σ2
−1− Σ2
+ − Σ2− −N2
+
, (20)
which is non-negative on BVI0 . Up to a constant multiple, this function is the same as Z−1
which was defined in [20] (a paper which quotes [3] as its source). The same references defined amarginally different monotone function, called Z1, in the case of Bianchi type VII0. This functionwas employed in [14, 8] to analyze Bianchi type VII0 asymptotic dynamics (which was in turn thecornerstone for the analysis of Bianchi type IX Mixmaster dynamics). In the following we use thefunction ζ as a building block in the analysis of Bianchi type VI0 asymptotics.
The condition ζ = 0 defines the orbit Γ. The condition ζ = z for R ∋ z > 0 represents (the surfaceof) a prolate spheroid in BVI0 ,
Σ2+ +
(1 + z−1
)Σ2
− +(1 + z−1
)N2
+ = 1 . (21)
This spheroid is embedded in the unit ball BVI0 in a characteristic way: First, the principal axisof the prolate spheroid coincides with the orbit Γ. Second, let Rs denote the radius of the discthat is orthogonal to the principal axis and arises as the intersection of the spheroid ζ = z withthe plane Σ+ = const; likewise, let Rd = (1 − Σ2
+)1/2 denote the radius of the disc that arises as
the intersection of the unit ball BVI0 with the plane Σ+ = const. Then Rs/Rd = (1 + 1/z)−1/2,i.e., the surface of the spheroid ζ = z is at a constant relative distance Rs/Rd from the centralorbit (axis) Γ.
While the function ζ is zero along the orbit Γ, ζ is positive and strictly monotonically decreasingalong every other orbit in BVI0 . To see this we use (17) to compute
ζ′ = −4Σ2
−(1 + Σ+)
1− Σ2+ − Σ2
− −N2+
≤ 0 (22a)
ζ′′|Σ−=0 = 0 (22b)
ζ′′′|Σ−=0 = −96(1 + Σ+)(1− Σ2
+)N2+
1− Σ2+ −N2
+
< 0 . (22c)
Remark. When calculating derivatives of ζ, it is convenient to use that
(1 − Σ2+ − Σ2
− −N2+)
′ = 4(Σ2+ +Σ2
− +Σ+)(1− Σ2+ − Σ2
− −N2+),
which follows straightforwardly from 1 − Σ2+ − Σ2
− − N2+ = − 1
3 N1N2, cf. (13) and (14), andfrom (9b).
The monotonicity of ζ can alternatively be expressed as follows: If ζ|γ(τ0) = ζ0 for an orbit γ(τ)at τ = τ0, then the orbit is contained within the spheroid ζ = ζ0 for all τ > τ0. The surface ζ = ζ0thus defines a ‘channel’ in which the orbit must be contained for all τ > τ0; this ‘channel’ directsthe orbit to T3, cf. Lemma 3.1.
4 Future asymptotics of Bianchi type VI0 vacuum models
The analysis of Section 3 suggests to introduce adapted variables. We define the variable
Σ+ = 1 + Σ+ , (23)
which is positive on BVI0 ; in addition, we introduce adapted ‘spheroidal coordinates’ ζ and ϑinstead of Σ− and N+ via
Σ− =
√1− Σ2
+√1 + ζ−1
cosϑ , (24a)
N+ = −
√1− Σ2
+√1 + ζ−1
sinϑ ; (24b)
4 FUTURE ASYMPTOTICS OF BIANCHI TYPE VI0 VACUUM MODELS 11
note that 1−Σ2+ = (2− Σ+)Σ+. The variable transformation (Σ+,Σ−, N+) 7→ (Σ+, ζ, ϑ) is one-to-
one on BVI0\Γ when we disregard translations by integer multiples of 2π in ϑ. Surfaces ζ = constrepresent the prolate spheroids; Σ+ = const its cross sections (orthogonal to the principal axis Γ);the variable ϑ is an angular variable in these sections.
Another beneficial change concerns the time variable: We introduce a new time variable σ through
d
dσ=
1
Σ+
d
dτ. (25)
By Lemma 3.1, for every orbit in BVI0 , Σ+ goes to zero as τ → ∞, where
Σ′+ = −2(1− Σ2
+ − Σ2−)Σ+,
see (17a). Consequently, 1− Σ2+ − Σ2
− /∈ L1([τ0,∞)) for every τ0 ∈ R. Since
0 < 1− Σ2+ − Σ2
− ≤ 1− Σ2+ = (2 + Σ+)Σ+ ≤ 2Σ+,
we infer that Σ+ /∈ L1([τ0,∞)) for every τ0 ∈ R. Since Σ+ is positive and
σ(τ) = σ(τ0) +
∫ τ
τ0
Σ+(s) ds,
we conclude that τ → ∞ corresponds to σ → ∞.
Using (Σ+, ζ, ϑ) as variables and σ as ‘time’, the dynamical system (17) becomes
dΣ+
dσ= −2Σ+(2− Σ+)
[1 + ζ sin2 ϑ
](1 + ζ)−1 , (26a)
dϑ
dσ= 2
√3 Σ
−1/2+ (2− Σ+)
1/2(1 + ζ sin2 ϑ)1/2(1 + ζ)−1/2 + sin 2ϑ , (26b)
dζ
dσ= −4ζ cos2 ϑ . (26c)
This system (together with the exceptional orbit Γ) completely describes the Bianchi type VI0dynamics. In the following, we present a detailed analysis of the system (26) and the properties ofits solutions.
Lemma 4.1. Consider a solution(Σ+(σ), ϑ(σ), ζ(σ)
)of the dynamical system (26) in BVI0\Γ.
Let σ0 ∈ R and let
g(σ) :=
∫ σ
σ0
cos2(ϑ(s)
)ds .
Then there exists a β > 0 such that
σ
2− β ≤ g(σ) ≤ σ
2+ β (27)
for all σ ≥ σ0.
Remark. Since log ζ(σ)−1/4 = log ζ(σ0)−1/4+g(σ) by (26c), the inequality (27) gives a first estimate
on the asymptotic behavior of ζ(σ) as σ → ∞, see Corollary 4.2.
Proof. Due to the fact that Σ+(σ) → 0 as σ → ∞ and the fact that ζ(σ) converges as σ → ∞, wecan assume σ0 to be large enough that there is a c0 > 1 such that
1
c0Σ
−1/2+ ≤ dϑ
dσ≤ c0Σ
−1/2+ (28)
for all σ ≥ σ0, cf. (26b). Using a simple trigonometric identity and changing variables leads to
g(σ) =1
2
σ∫
σ0
[1 + cos
(2ϑ(s)
)]ds =
1
2(σ − σ0) +
1
2
ϑ(σ)∫
ϑ(σ0)
cos(2θ)(dϑdσ
)−1
dθ . (29)
4 FUTURE ASYMPTOTICS OF BIANCHI TYPE VI0 VACUUM MODELS 12
Inserting (28) we obtain
1
2c0
ϑ(σ)∫
ϑ(σ0)
cos(2θ) Σ1/2+ dθ ≤ g(σ)− 1
2(σ − σ0) ≤ c0
2
ϑ(σ)∫
ϑ(σ0)
cos(2θ) Σ1/2+ dθ . (30)
Here we consider Σ+ to be a function of ϑ, where we have
dΣ+
dϑ=
dΣ+
dσ
(dϑdσ
)−1
< 0 ,
assuming σ to be large enough, cf. (28). Let us estimate the integral in (30).
If 2θn = −π/2 + 2nπ, with n ∈ N sufficiently large, then
∫ θn+1
θn
Σ1/2+ cos(2θ) dθ ≥ 0 . (31)
If it were not for the factor Σ1/2+ , the integral would be zero. The reason for the inequality (31) is
that the infimum of this factor over the subset of the interval of integration where the integrandis positive equals the supremum of this factor over the subset of the interval of integration wherethe integrand is negative. For similar reasons, if 2θn = π/2 + 2nπ, n ∈ N, then
∫ θn+1
θn
Σ1/2+ cos(2θ) dθ ≤ 0 .
Therefore, since the integral ∫ ϑ(σ)
ϑ(σ0)
Σ1/2+ cos(2θ) dθ
can be written as a sum of non-negative integrals or as a sum of non-positive integrals, up to anerror corresponding to, at worst, one interval at the beginning of the interval of integration andone interval at the end of the interval of integration, we conclude that
∫ ϑ(σ)
ϑ(σ0)
Σ1/2+ cos(2θ)dθ = O
(Σ
1/2+ (σ0)
).
Since this estimate is uniform in σ, we are led to the conclusion that
∫ ϑ(σ)
ϑ(σ0)
Σ1/2+ cos(2θ)dθ
converges as σ → ∞, and that there is a constant, say A, such that
∫ ϑ(σ)
ϑ(σ0)
Σ1/2+ cos(2θ) dθ = A+O
(Σ
1/2+ (σ)
). (32)
In particular, the conclusions of the lemma follow.
Corollary 4.2. Along every orbit in BVI0 we have ζ → 0 as σ → ∞ (or τ → ∞).
Proof. For the orbit Γ the statement is trivial. For every other orbit we use Lemma 4.1. Sincelog ζ(σ)−1/4 = log ζ(σ0)
−1/4 + g(σ), Lemma 4.1 implies that there exists a c1 > 1 such that
c−11 e−2σ ≤ ζ(σ) ≤ c1e
−2σ (33)
for all σ ≥ σ0.
4 FUTURE ASYMPTOTICS OF BIANCHI TYPE VI0 VACUUM MODELS 13
Lemma 4.1, in combination with the fact that Σ+ → 0 (σ → ∞), suggests that the asymptoticbehavior of solutions of (26) is approximately determined by the system
dΣ+/dσ = −4Σ+ , (34a)
dϑ/dσ = 2√6 Σ
−1/2+ , (34b)
dζ/dσ = −2ζ . (34c)
This statement is made precise in the following lemma.
Lemma 4.3. Consider a solution(Σ+(σ), ϑ(σ), ζ(σ)
)of the dynamical system (26) in BVI0\Γ.
Then there are positive constants a, b =√6/a, and c such that
Σ+(σ) = a e−4σ[1 + O(e−2σ)
], (35a)
ϑ(σ) = b e+2σ[1 +O(σe−2σ)
], (35b)
ζ(σ) = c e−2σ[1 +O(e−2σ)
](σ → ∞) . (35c)
Proof. Due to Lemma 4.1, see also (33), we have ζ = O(e−2σ). Eq. (26a) thus reads
dΣ+/dσ = −2Σ+(2− Σ+)[1 +O(e−2σ)
],
whence Σ+ is (at least) of order O(e−2σ), since Σ+ → 0 (σ → ∞). Reinserting this informationinto (26a), we thus have
dΣ+/dσ = −4Σ+
[1 +O(e−2σ)
],
so thatΣ+(σ) = exp[−4σ +A+O(e−2σ)]
for some constant A; Eq. (35a) follows. Combining this observation with (26b), we conclude that
dϑ/dσ = 2√6 Σ
−1/2+
[1 +O(e−2σ)
],
and (35b) ensues by a simple integration. Finally, consider
g(σ)− 1
2(σ − σ0) =
1
2
ϑ(σ)∫
ϑ(σ0)
cos(2θ)(dϑdσ
)−1
dθ =1
4√6
ϑ(σ)∫
ϑ(σ0)
cos(2θ) Σ1/2+
[1 +O(e−2σ(θ))
]dθ
=1
4√6
ϑ(σ)∫
ϑ(σ0)
cos(2θ) Σ1/2+ dθ +
1
2
σ∫
σ0
cos(2ϑ(σ)
)O(e−2σ)
[1 +O(e−2σ)
]dσ , (36)
where we have changed back the integration variable in the second integral. Due to (32), we have
∫ ϑ(σ)
ϑ(σ0)
1
4√6Σ
1/2+ cos(2θ) dθ = B +O
(Σ
1/2+ (σ)
)= B +O(e−2σ),
for some constant B, where we have used the fact that Σ+(σ) = O(e−4σ). The second integralin (36) is of the form C +O(e−2σ). To conclude,
g(σ) =1
2σ +D +O(e−2σ) ,
and (35c) follows.
Lemma 4.4. Consider a solution in BVI0\Γ. Then there is a ζ0 > 0 such that
Σ+(τ) =14τ
−1 [1 +O(τ−1/2)] , (37a)
ϑ(τ) = 2√6 τ1/2 +O(log τ) , (37b)
ζ(τ) = ζ0τ−1/2 [1 +O(τ−1/2)] , (37c)
as τ → ∞.
5 COMPARISON WITH TYPE II DYNAMICS 14
Proof. Using (25) and (35a), we find that
σ =1
4log τ +
1
4log(4a) +O(τ−1/2) (τ → ∞) .
Thus the statements of the lemma are essentially immediate consequences of observations alreadymade.
It follows from Lemma 4.4 that there exists a positive constant r0 such that
√1− Σ2
+(τ)√1 + ζ−1(τ)
= r0τ−3/4 [1 +O(τ−1/2)]
as τ → ∞. Therefore, we finally obtain
Proposition 4.5. Consider a solution in BVI0\Γ. Then there is an r0 > 0 such that the asymptoticdynamics are given by
Σ+ = −1 + 14τ
−1[1 +O(τ−1/2)
], (38a)
Σ− = r0τ−3/4 cosϑ(τ)
[1 +O(τ−1/2)
], (38b)
N+ = −r0τ−3/4 sinϑ(τ)
[1 +O(τ−1/2)
], (38c)
N− = 1√2τ−1/2
[1 +O(τ−1/2)
]. (38d)
Remark. Every solution in BVI0 has an asymptotic representation of the form (38); for the orbitΓ, we set r0 = 0. (In the case of Γ, a simplified analysis in fact leads to somewhat better errorestimates.) Note that there is merely one free parameter in (38) — at this level of approximation,it is a one-parameter family of orbits in BVI0 that has the same leading asymptotics, cf. Figure 3.
Remark. From the results of Proposition 4.5, the asymptotic behavior of the variables (Σ1,Σ2,Σ3)and (N1, N2) follows directly. It is thus straightforward to obtain the asymptotic expressions forthe metric via the transformations of Section 2.
5 Comparison with type II dynamics
In this section, we discuss one important consequence of the result (38): Type VI0 future asymptoticdynamics are totally unrelated to type II dynamics. Let us elaborate.
Consider a type II orbit on the hemisphere B± of ∂BVI0 , see Figure 1. When projected ontothe (Σ+,Σ−)-plane, such an orbit is represented by a straight line Σ+ ± nΣ− = −1 +
√3n with
n ∈ [0,√3]; n = 0 corresponds to the point T3, n = 1/
√3 to the orbit T1 7→ Q1 [T2 7→ Q2], and
n =√3 to the point T2 [T1] in the case of the northern [southern] hemisphere; see Figure 2 and,
e.g., [19] for further details. Consequently, a type II orbit on B± is the intersection of the planeΣ+ ± nΣ− = −1 +
√3n in R3 with B±.
A good representation of the structure of the flow on ∂BVI0 is obtained by projecting the type IIorbits onto the (Σ−, N+)-plane. Let n be small, i.e., we restrict our attention to orbits in thevicinity of the Taub point T3. In that case, Σ−/N+ is monotonically increasing and exhauststhe interval (−∞,∞). Hence we can parametrize an orbit on B± close to T3 by θ in sucha way that Σ−/N+ = − cot θ, where θ has the range (−π, 0) and (0, π) for the northern andsouthern hemisphere, respectively; accordingly, the orbit is represented by the parametric curve(Σ+,Σ−, N+) =
(√1− r(θ)2, r(θ) cos θ,−r(θ) sin θ
). Using Σ+ ± nΣ− = −1 +
√3n and the rela-
tion Σ− = r(θ) cos θ, one obtains a second order equation for r(θ). This equation has the (positive)solution
r(θ) = (2√3)1/2n1/2 ∓ n cos θ +O(n3/2)
5 COMPARISON WITH TYPE II DYNAMICS 15
Figure 3: This figure shows the part Σ+ < −1/3 of the state space BVI0 . The outer shell is ∂BVI0 ,which can be represented as (Σ2
− + N2+)
1/2 = (1 − Σ2+)
1/2; in the vicinity of T3 this corresponds
to (Σ2− + N2
+)1/2 ≃
√2(1 + Σ+)
1/2. The two ‘cones’ are surfaces (Σ2− + N2
+)1/2 = a(1 + Σ+)
3/4
for two different values of the constant a. By (38), a type VI0 orbit that is contained on such acone initially remains (approximately) on that cone for all later times; in fact, asymptotically, atype VI0 orbit spirals in towards T3 along such a ‘cone’.
for small n. Introducing c0 = (2√3n)−1/2, this can be expressed
r(θ) =(c0 ±
1
2√3cos θ
)−1
+O(c−30 ) . (39)
In practice, we will ignore the lower order terms in what follows.
Concatenating type II orbits (in the natural manner so that the ω-limit point of orbit numberk − 1 coincides with the α-limit point of orbit number k), we obtain a sequence of type II orbits.Using (39) we see that the projection of this sequence onto the (Σ−, N+)-plane is represented bythe spiral (Σ−, N+) = r(θ)(cos θ,− sin θ) with θ > θ0 for some θ0 ∈ R and
r(θ) =2√3
c1 + cs(θ), (40)
where cs(θ) is defined as∫ θ
0| sin(s)|ds and c1 is a (large) constant. Since cs(θ) = (2/π) θ [1+O(θ−1)],
an asymptotic representation of this spiral is
r(θ) =
√3 π
θ[1 +O(θ−1)] . (41)
This ‘type II spiral’ is depicted in Figure 4(a).
The Kasner parameter, u, is a parameter that (uniquely) characterizes the fixed points in a vicinityof the Taub point T3 (which is itself represented by u = ∞); see, e.g., [19] for details. Theconnection between the Kasner parameter u and the radius r is very simple: u =
√3 r−1[1+O(r−1)]
5 COMPARISON WITH TYPE II DYNAMICS 16
-0.1 0.1Σ−
N+
(a) Projection of type II orbits
-0.1 0.1Σ−
N+
(b) Projection of a BVI0 orbit
Figure 4: Subfigure (a) depicts the projection of the flow on ∂BVI0 (in a vicinity of the Taubpoint T3) onto (Σ−, N+)-space. Each hemicircle is the projection of one type II orbit; the lineN+ = 0 is the projection of the Kasner circle. Concatenating type II orbits in the natural wayresults in a spiral, see (41). Subfigure (b) depicts the projection of an actual orbit in BVI0 onto(Σ−, N+)-space. The resulting spiral, which is described by (43), is completely different from the‘type II spiral’ in subfigure (a) — the asymptotic evolution of type VI0 models is unrelated totype II dynamics.
(see, e.g., [7]). Consequently, setting θ = kπ with k ∈ N (so that k numbers the type II orbits ofthe sequence of orbits), Eq. (41) leads to2
uk = u∣∣θ=kπ
=
√3
r(kπ)= k + u0 . (42)
This is the well-known result for a sequence of type II orbits (Kasner sequence), which correspondsto the k-fold iteration of the Kasner map u 7→ u+ 1 with initial value u0.
The type VI0 dynamics are different. Consider an arbitrary orbit in BVI0 . The asymptoticdynamics of this orbit are given by Proposition 4.5. From (37b) and (38), with ρ0 = 243/4r0,we thus obtain (Σ−, N+) = r(ϑ)(cos ϑ,− sinϑ) with
r(ϑ) = ρ0ϑ−3/2
[1 +O(ϑ−1 logϑ)
]. (43)
This represents a spiral in the (Σ−, N+)-plane which is completely different from the type IIspiral (41); see Figure 4(b). Setting ϑ = kπ, k ∈ N, leads to r(kπ) ∝ k−3/2 [1+O(k−1 log k)]; whenwe further define uk, in analogy with the above, by uk =
√3/r(kπ), then
uk = u0 k3/2
[1 +O(k−1 log k)
]. (44)
Hence the asymptotic evolution of the (analog of the) Kasner parameter is radically different fromthe type II (Kasner map) evolution.
Remark. A solution in BVI0 whose initial data is chosen extremely close to ∂BVI0 shadows asequence of type II orbits for some time. Accordingly, such a solution behaves like (41), cf. Fig-ure 4(a), for some time. However, eventually, as proved by Proposition 4.5, the solution deviatesfrom this type of evolution and behaves asymptotically like (43), cf. Figure 4(b).
Remark. A different way of illustrating the differences between type II and type VI0 evolution isto track the key quantity |N1/N2|. For every type VI0 orbit we obtain from (38) that
∣∣∣N1
N2
∣∣∣ = 1− n0τ−1/4 sinϑ(τ) +O(τ−1/2) (45)
2The O(·)-term yields a constant for general reasons.
REFERENCES 17
as τ → ∞ for some constant n0. Hence, |N1/N2| oscillates around 1, but the amplitude ofthe oscillations converges to zero as τ → ∞. In contrast, if a type VI0 were asymptotic to asequence of type II orbits, |N1/N2| would oscillate with growing amplitudes: lim inf |N1/N2| = 0and lim sup |N1/N2| = ∞.
By the results of [14], see also [8, 9], it is known, in a sense that has to remain vague here, thatBianchi type IX solutions are approximated by sequences of type II solution as the singularityis approached. For Bianchi type VIII it is unclear whether this result holds analogously, cf. alsothe remarks in the introduction. It will be the role type VI0 plays in the context of type VIIIdynamics that will decide whether singularities associated with Bianchi type VIII are local andwhether they are oscillatory in the same sense (type II oscillations) as singularities of type IX.In any case, answering these questions will have broad ramifications for our understanding of thenature of generic singularities.
Acknowledgments
We gratefully acknowledge the hospitality of the Mittag-Leffler Institute, which was afforded usduring the programme entitled “Geometry, Analysis, and General Relativity”. H. R. is supportedthe Goran Gustafsson Foundation and is a Royal Swedish Academy of Sciences Research Fellowsupported by a grant from the Knut and Alice Wallenberg Foundation.
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